Some applications of the condensation of the singularities of families of nonnegative functions

RecentlyPrus-Wi?niowski [14] has proved that the continuous functions of Λ-bounded variation on [0, 1] form a set of the first category in the Banach space C[0, 1] and also in each Banach space CΓBV[0, 1] of continuous functions of Γ-bounded variation on [0, 1] provided that the sequence Γ is adequate. In the present paper these results are extended and completed by using a principle of the condensation of the singularities of a family of nonnegative functions that follows from the theorems given byBreckner [3]. It is shown that Baire category properties similar to those stated in [14] are valid for two very large classes of real-valued functions called functions of bounded λ-variation of orderp and functions of boundedmth variation of orderp, respectively. The benefit of considering these classes is that they comprise several classes of functions of bounded variation type which have occurred so far in Fourier analysis or real analysis; in particular, the functions investigated in [14]. Thus, by specializing the results derived in the present paper, they give at once Baire category information concerning a number of well-known sets of real-valued functions.     

On the continuity of superposition operators in the space of functions of bounded variation

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).     

Trace theorems for functions of bounded variation in metric spaces

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality, and obtain $L^1$ estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'ya-type inequality for functions of bounded variation that vanish on a set of positive capacity.     

Epigraphical and Uniform Convergence of Convex Functions

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of .

     

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

Acharacterization of Newtonian functions with zero boundary values

We give an intrinsic characterization of the property that the zero extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.     

Approximation of lipschitz functions by Δ-convex functions in banach spaces

In this paper we give some result about the approximation of a Lipschitz function on a Banach space by means of Δ-convex functions. In particular, we prove that the density of Δ-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of Δ-convex functions.     

Hypersurfaces of prescribed mean curvature in central projection-I

We consider the Dirichlet-problem associated with H-surfaces in central projection. By using a variational approach in the space of functions of bounded variation, and in connection with the theory of minimal sets, we obtain very general existence and regularity results provided the mean curvature H fulfils a certain monotony condition. Received: 11 February 2003     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

On the strong CE-property of convex sets

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.     

Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.     

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

Approximation in variation for Mellin integral operators

We study approximation results for a family of Mellin integral operators in the space of functions of bounded variation in the multidimensional setting, by means of a new concept of variation for multivariate functions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parallel Transportation for Alexandrov Space with Curvature Bounded Below

In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula for this case. A closely related construction has been made for Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).?Naturally, as we have a more general situation, the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular, we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.?Author is indebted to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks. Submitted: January 1997, Revised version: June 1997     

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin-Kravchenko-Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S¹, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S¹-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111-127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.